How do I play animations in the game engine?
This can be done in the game logic view using an action actuator. First ensure the Armature Modifier on your mesh is at the top of the Modifier stack or at least above Subserf.
Change the screen layout to Game logic and the physics type to Blender Game using the drop-downs at the top of the screen.



Select your object that you want the animation to play from and in the logic Editor space, (large grey empty bottom of the screen) click the Add Sensor button and select always. 



Now remember that a sensor sets conditions for something to happen, so you may not want to use the Always in different circumstances. 

Click Add Controller > AND along with Add Actuator > Action.

Select your animation in the light grey box on the action actuator 



and set the start and end values to encompass the entire length of the animation. My animation is 60 frames long, so I set start to 1 and end to 60.



The last step is to connect all the little dots in between the boxes so that it forms one string of logic. Always > AND > Action



Press P to play the game and see the animation play.